Program extraction for 2-random reals

Abstract

Let \({2-\textsf{RAN}}\) be the statement that for each real X a real 2-random relative to X exists. We apply program extraction techniques we developed in Kreuzer and Kohlenbach (J. Symb. Log. 77(3):853–895, 2012. doi:10.2178/jsl/1344862165), Kreuzer (Notre Dame J. Formal Log. 53(2):245–265, 2012. doi:10.1215/00294527-1715716) to this principle. Let \({{\textsf{WKL}_0^\omega}}\) be the finite type extension of \({\textsf{WKL}_0}\). We obtain that one can extract primitive recursive realizers from proofs in \({{\textsf{WKL}_0^\omega} + \Pi^0_1-{\textsf{CP}} + 2-\textsf{RAN}}\), i.e., if \({{\textsf{WKL}_0^\omega} + \Pi^0_1-{\textsf{CP}} + 2-\textsf{RAN} \, {\vdash} \, \forall{f}\, {\exists}{x} A_{qf}(f,x)}\) then one can extract from the proof a primitive recursive term t(f) such that \({A_{qf}(f,t(f))}\). As a consequence, we obtain that \({{\textsf{WKL}_0}+ \Pi^0_1 - {\textsf{CP}} + 2-\textsf{RAN}}\) is \({\Pi^0_3}\) -conservative over \({\textsf{RCA}_0}\).